The acoustic performance of manufactured products is becoming an extremely important aspect in the design and development process, not only to improve the comfort of the user of the product (e.g. passengers in a car, in an aircraft), but also to reduce the nuisance to the surroundings (e.g. habitations close to highways or to an airport). A typical problem may be described or represented for numerical analysis as a vibrating body surrounded by a fluid which may be assumed effectively infinite at least as an approximation.
The concept of Acoustic Transfer Vectors (ATV's) is known [Y. K. Zhang, M.-R. Lee, P. J. Stanecki, G. M. Brown, T. E. Allen, J. W. Forbes, Z. H. Jia, “Vehicle Noise and Weight Reduction Using Panel Acoustic Contribution Analysis”, SAE95, Paper 95NV69].
ATV's are input-output relations between the normal structural velocity of the vibrating surface and the sound pressure level at a specific field point (see FIG. 3). ATV's can be interpreted as an ensemble of Acoustic Transfer Functions from the surface nodes to a single field point. As such ATV's can be measured and are a physical parameter of a system. Literature refers sometimes to this concept as Contribution Vectors or Acoustic Sensitivities.
ATV's only depend on the configuration of the acoustic domain, i.e. the shape of the vibrating body and the fluid properties controlling the sound propagation (speed of sound and density), the acoustic surface treatment, the frequency and the field point. They do not depend on the loading condition. The concept of ATV's, and their properties, is not new, and has already been published in several scientific papers.
There are two computer-aided engineering processes that are key for evaluating and optimising the acoustic performance of structures: the ability to predict the acoustic radiation pattern of a vibrating structure, either from computed or measured surface vibrations (acoustic radiation prediction), and the ability to recover surface vibrations onto a vibrating structure from measured field sound pressure level. The latter one is sometimes the only manner to perform source identification, when it is impossible to apply measurement devices onto the structure surface (e.g. rotating tire).
These engineering processes rely on numerical analysis methods, amongst which the most popular are currently the various forms of the boundary element method (the direct mono-domain, the direct multi-domain and the indirect approaches), and the finite element method, that can be extended to handle unbounded regions, e.g. using infinite elements. Numerical analysis is well known to the skilled person, e.g. “The finite element method”, Zienkiewicz and Taylor, Butterworth-Heinemann, 2000; “Numerical Analysis”, Burden and Faires, Brooks/Cole, 2001; H. A. Schenck, “Improved integral formulation for acoustics radiation problems”, J. Acoust. Soc. Am., 44, 41-58 (1981); A. D. Pierce, “Variational formulations in acoustic radiation and scattering”, Physical Acoustics, XXII, 1993.; J. P. Coyette, K. R. Fyfe, “Solutions of elasto-acoustic problems using a variational finite element/boundary element technique”, Proc. of Winter Annual Meeting of ASME, San Francisco, 15-25 (1989).
U.S. Pat. Nos. 5,604,893 and 5,604,891 describe finite element methods for solving acoustic problems by using an oblate finite element. The oblate finite element is based on a multipole expansion that describes a scattered or radiated field exterior to an oblate spheroid.
The computational cost associated with numerical methods for acoustic performance prediction is usually very large, and is linearly proportional to:                the number of operational conditions (may be about a 100 different cases);        the number of frequency lines to be evaluated for obtaining a representative response function (will typically be about 100 to 200);        the number of design variants to be studied.        